# Embedded Cobordism Categories and Spaces of Submanifolds

@article{RandalWilliams2010EmbeddedCC, title={Embedded Cobordism Categories and Spaces of Submanifolds}, author={Oscar Randal-Williams}, journal={International Mathematics Research Notices}, year={2010}, volume={2011}, pages={572-608} }

Galatius, Madsen, Tillmann, and Weiss [7] have identified the homotopy type of the classifying space of the cobordism category with objects (d −1)-dimensional manifolds embedded in ℝ ∞ . In this paper we apply the techniques of spaces of manifolds, as developed by the author and Galatius in [8], to identify the homotopy type of the cobordism category with objects (d −1)-dimensional submanifolds of a fixed background manifold M. There is a description in terms of a space of sections of a bundle…

## 12 Citations

The homotopy type of the topological cobordism category

- Mathematics
- 2018

We define a cobordism category of topological manifolds and prove that if $d \neq 4$ its classifying space is weakly equivalent to $\Omega^{\infty -1} MTTop(d)$, where $MTTop(d)$ is the Thom spectrum…

A metric for the space of submanifolds of Galatius and Randal-Williams

- Mathematics
- 2015

Galatius and Randal-Williams defined a topology on the set of closed submanifolds of ${\mathbb R}^n$. B\"okstedt and Madsen proved that a $C^1$-version of this topology is metrizable by showing that…

A relative h-principle via cobordism-like categories

- Mathematics
- 2010

We prove an h-principle with boundary condition for a certain class of topological spaces valued sheaves. The techniques used in the proof come from the study of the homotopy type of the cobordism…

Parametrized geometric cobordism and smooth Thom stacks

- Mathematics
- 2017

We develop a theory of parametrized geometric cobordism by introducing smooth Thom stacks. This requires identifying and constructing a smooth representative of the Thom functor acting on vector…

A T ] 3 S ep 2 01 7 Parametrized geometric cobordism and smooth Thom stacks

- Mathematics
- 2018

We develop a theory of parametrized geometric cobordism by introducing smooth Thom stacks. This requires identifying and constructing a smooth representative of the Thom functor acting on vector…

Tori detect invertibility of topological field theories

- MathematicsGeometry & Topology
- 2018

A once-extended d-dimensional topological field theory Z is a symmetric monoidal functor (taking values in a chosen target symmetric monoidal (infty,2)-category) assigning values to (d-2)-manifolds,…

Homotopy types of spaces of submanifolds of ${\mathbb R}^n$

- Mathematics
- 2014

We compute the homotopy type of the space of proper d-dimensional submanifolds of ${\mathbb R}^n$ with a smooth version of the Fell topology. Our methods allow us to compute the homotopy type of the…

Stable moduli spaces of high‐dimensional handlebodies

- Mathematics
- 2015

We study the moduli space of handlebodies diffeomorphic to (Dn+1×Sn)♮g , that is, the classifying space BDiff((Dn+1×Sn)♮g,D2n) of the group of diffeomorphisms that restrict to the identity near an…

The space of merging submanifolds in R^n

- Mathematics
- 2014

We compute the homotopy type of the space of possibly empty proper $d$-dimensional submanifolds of R^n with a topology coming from a Hausdorff distance. Our methods give also a different proof of the…

Invertible Topological Field Theories

- Mathematics
- 2017

A $d$-dimensional invertible topological field theory is a functor from the symmetric monoidal $(\infty,n)$-category of $d$-bordisms (embedded into $\mathbb{R}^\infty$ and equipped with a tangential…

## References

SHOWING 1-10 OF 18 REFERENCES

Geometric cobordism categories

- Mathematics
- 2009

In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections…

The homotopy type of the cobordism category

- Mathematics
- 2006

The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main…

Monoids of moduli spaces of manifolds

- Mathematics
- 2010

We study categories of d-dimensional cobordisms from the perspective of Tillmann and Galatius-Madsen-Tillmann-Weiss. There is a category $C_\theta$ of closed smooth (d-1)-manifolds and smooth…

STABLE CHARACTERISTIC CLASSES OF SMOOTH MANIFOLD BUNDLES

- Mathematics
- 2009

Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union t BSOn of classifying spaces of special orthogonal groups SOn with n = 0,1, . . .. A…

Embeddings from the point of view of immersion theory

- Philosophy
- 1999

Let M and N be smooth manifolds. For an open V ⊂ M let emb(V,N) be the space of embeddings from V to N . By the results of Goodwillie [4], [5], [6] and Goodwillie–Klein [7], the cofunctor V 7→…

Homotopy Limits, Completions and Localizations

- Mathematics
- 1987

Completions and localizations.- The R-completion of a space.- Fibre lemmas.- Tower lemmas.- An R-completion of groups and its relation to the R-completion of spaces.- R-localizations of nilpotent…

Configuration-spaces and iterated loop-spaces

- Mathematics
- 1973

The object of this paper is to prove a theorem relating "configurationspaces" to iterated loop-spaces. The idea of the connection between them seems to be due to Boardman and Vogt [2]. Part of the…

Embeddings from the point of view of immersion theory: Part II

- Mathematics
- 1999

Let M and N be smooth manifolds without boundary. Immersion theory suggests that an understanding of the space of smooth embeddings emb(M,N) should come from an analysis of the cofunctor V |-->…

A parametrized index theorem for the algebraicK-theory Euler class

- Mathematics
- 2003

A Riemann–Roch theorem is a theorem which asserts that some algebraically defined wrong–way map in K –theory agrees or is compatible with a topologically defined one [BFM]. Bismut and Lott [BiLo]…